mathematics Article Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs Josephine Brooks 1, Alvaro Carbonero 2, Joseph Vargas 3 , Rigoberto Flórez 4, Brendan Rooney 5 and Darren Narayan 5,* 1 Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada; [email protected] 2 Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154-4020, USA; [email protected] 3 Mathematical Sciences Department, State University of New York, Fredonia, NY 14063, USA; [email protected] 4 Department of Mathematical Sciences, The Citadel, Charleston, SC 29409, USA; rfl[email protected] 5 School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA; [email protected] * Correspondence: [email protected] Abstract: An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence Citation: Brooks, J.; Carbonero, A.; graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present Vargas, J.; Flórez, R.; Rooney, B.; a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, Narayan, D. Removing Symmetry in thus removing all symmetries from the graph. Circulant Graphs and Point-Block Incidence Graphs. Mathematics 2021, Keywords: fixing number; circulant graph; point-block incidence graph; asymmetric graph 9, 166. https://doi.org/10.3390/ math9020166 Received: 14 December 2020 1. Introduction Accepted: 8 January 2021 If you look closely at a QR-code you will see that three of the corners are marked Published: 14 January 2021 with small nested squares. Marking these corners gives the QR-code a fixed orientation Publisher’s Note: MDPI stays neu- (where the original, and rotations of 90, 180 and 270 degrees are all different). The marked tral with regard to jurisdictional clai- corner adjacent to two other marked corners is placed in the upper left. With this fixed ms in published maps and institutio- orientation a smart phone can read it and know how it should be oriented accordingly. nal affiliations. QR-codes are designed in black and white; however, if they included an additional colour, say blue, we would only need to mark two adjacent corners, one with a blue nested square and one with a black nested square. The blue square could correspond to the upper left corner and the black square could correspond to the upper right corner. We note that it is Copyright: © 2021 by the authors. Li- possible to fix two corners in black and white if the nested squares are drawn differently. censee MDPI, Basel, Switzerland. This QR-code example shows that the fixing number of C4 is 2. This article is an open access article In this paper, we consider the problem of removing symmetry from a graph by distributed under the terms and con- fixing vertices. We say that a vertex v in a graph G is fixed if it is mapped to itself under ditions of the Creative Commons At- every automorphism of G. The fixing number of a graph G is the minimum number tribution (CC BY) license (https:// of vertices, when fixed, fixes all of the vertices in G, and as a result all symmetries of creativecommons.org/licenses/by/ 4.0/). the graph are removed. The determination of fixing numbers are important as they Mathematics 2021, 9, 166. https://doi.org/10.3390/math9020166 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 166 2 of 16 provide insight into the famous problem of determining the automorphism group of a graph. Fixing numbers were introduced by Gibbons and Laison [1], and independently by Erwin and Harary [2]. Fixing numbers have also been called determining numbers by Boutin [3]. Fixing/determining numbers have been investigated for many families of graphs including complete graphs, paths, cycles [2], Cayley graphs and Frucht graphs [1], Cartesian products [4] and Kneser graphs [5]. Recently fixing numbers were determined for cographs and unit interval graphs [6]. We also note a concrete application as the removing of symmetry from graphs is related to a problem in robotic manipulation where the goal is to determine the orientation of a marked sphere from a single visual image [7]. In this paper, we investigate circulant graphs and point-block incidence graphs which arise from combinatorial designs. In both families, the graphs are regular and their circular structures are closely tied to modular arithmetic. We will use fix(G) to denote the fixing number of a graph G. We will refer to a vertex v 2 V(G) as distinguishable if it is fixed under every automorphism of G. Following the definition from in [8] two vertices u and v are twins if they have the same open neighbourhoods or the same closed neighbourhoods. For any undefined notation, please see the text by West [9]. In this paper, we determine fixing numbers for multiple families of graphs. In Section2 , we identify the fixing number for powers of paths, which motivates the results in Section3, including a similar result for powers of cycles. This leads us to an investigation of fixing numbers of circulant graphs. In Section4, we investigate fixing numbers for point-block incidence graphs. This includes presentation of infinite families of graphs where fixing any vertex fixes every vertex, thus removing all symmetries from the graph. In the Conclusion section we pose ideas for future study. 2. Powers of Paths m We begin by determining the fixing number of powers of paths. The graph Pn (m ≤ n) is the graph with v1, v2, ..., vn as its set of vertices and fvivj : ji − jj ≤ mg as its set of edges. A small example is shown in Figure1. v1 v2 v3 v4 v5 3 Figure 1. The graph P5 . To obtain the fixing number, we first need to examine the degrees of the vertices in a power of a path. A vertex that is adjacent to all other vertices will be called a spanning vertex. We note that a graph with no non-trivial automorphisms can have at most one spanning vertex. A graph with two spanning vertices will have an automorphism that transposes the two spanning vertices. n m Lemma 1. If m ≥ 2 , then Pn has 2m − n + 2 spanning vertices. Furthermore, for every i where 1 ≤ i ≤ n − m − 1, there exist exactly two vertices with degree n − 1 − i. Proof. Following the definition, we notice that the vertices vi where i 2 [n − m, m + 1] have the property that ji − jj ≤ m for every j 2 [1, n]. By definition, this implies that these vertices have degree n − 1. Thus, there are m + 1 − (n − m) + 1 = 2m − n + 2 spanning vertices. Notice that vn−m−1 and vm+2 have degree n − 2. Similarly, vn−m−2 and vm+3 have degree n − 3. In general, vn−m−i and vm+1+i have degree n − 1 − i, thus completing the proof. Theorem 1. Let n and m be integers such that n > m. m 1. If m = n − 1, then fix(Pn ) = n − 1. Mathematics 2021, 9, 166 3 of 16 n m 2. If n − 2 ≥ m ≥ 2 , then fix(Pn ) = 2m − n + 2 n m 3. If 2 > m, then fix(Pn ) = 1. Proof. The first case is a complete graph, therefore fix(Kn) = n − 1. n We now prove the second case. As m ≥ 2 , then there exists 2m − n + 2 spanning m vertices in Pn . We must distinguish 2m − n + 2 − 1 of them (all but one). Let Vi = fvi, vn−i+1g for each i 2 [1, n − m − 1]. Notice that as automorphisms preserve degrees, every automorphism maps Vi to itself for each i. We break these symmetries by fixing v1. Then, vn is fixed and of the non-spanning vertices, half of the vertices vi will be adjacent to v1 and the other half vn−i will not be adjacent to v1. Therefore, there is no automorphism that will swap vi with vn−i. It follows that the only automorphism is the trivial one. As we removed all symmetries by fixing only one non-spanning vertex, we conclude the fixing number is n − 2k + 2 − 1 + 1 = n − 2k + 2. For the third case we note the graph has only one non-trivial automorphism, where vi n is swapped with vn−i+1 for all 1 ≤ i ≤ 2 . When n is even we can fix any vertex and remove this automorphism, when n is odd we can fix any vertex other than the centre vertex and remove this automorphism. 3. Fixing Numbers of Circulant Graphs The circulant graph Cn A, where A ⊆ [n], is the graph with set of vertices equal to Zn and set of edges equal to fij : (i − j) mod n 2 A or (j − i) mod n 2 Ag.